The proof is similar to that of [1]}. We first show that the functor is fully faithful. Let \((g,X), (g^{\prime },X^{\prime })\in \mathbf {B}^{\varphi ,\nabla }(G,R)\) , then any morphism \(\eta \colon I(g,x) \rightarrow I(g^{\prime },X^{\prime })\)  is an isomorphism, since any morphism of tensor functors between rigid tensor categories is an isomorphism by [2]}. By composing \(\eta \)  with the forgetful functor, we then have an automorphism of the fiber functor \(\omega ^G\otimes R\) . By Corollary REF , this automorphism is given by a unique element \(x\in G(R)\) , which then gives an isomorphism between \((g,X)\)  and \((g^{\prime },X^{\prime })\) , as desired.
